A Simple Derivation of the Equations of Einstein's
Theory of Special Relativity
P.S.C. Bruskiewich
Mathematical-Physics, University of British Columbia, Vancouver, BC
The implications of the two simple postulates of Special Relativity proposed by Albert
Einstein are presented in a succinct form in this paper. The simple algebraic derivation
of the equations of Special Relativity was shared by the author to Dr. F. A. Kaempffer
and Dr. George Volkoff during the course of a pleasant conversation in 1984.
1.0 Einstein's two Simple Postulates of Special Relativity
As proposed by Albert Einstein, there are two simple postulates in Special Relativity
i) There is no preferred inertial frame,
ii) The fasted a signal can travel in a frame of reference is the speed of light
The implications of these two simple postulates is the Lorentz transformation which can
be easily derived using simple algebra.
2.0 The Lorentz Transformation
Consider a transformation from a rest frame to a moving frame (the moving frame is
denoted by prime),
x = /(jc' + vf')
x = y(x-vt)
Evidentially if one frame is moving to the right with respect to the other, the other frame
is moving to the left with respect to the first frame of reference.
If a light beam is moving in both frames this
x-ct
x' = ct f
From this we find
ct = y (ct' + vf') =^> ct = y (c + v) t'
ct' = y (ct - vt) ^> ct' = y (c - v) t
If we multiply the two right hand equations together we find
ct(ct') = y (c + v)t'y(c-v)t^>c 2 =^ 2 (c 2 -v 2 )
From whence we derive the Lorentz Transformation
7 =
1
'1-
The two equations then become
V
c
x
x
(x' + vf)
(x-vtj
3.0 Einstein's Third Postulate of Special Relativity
Notice how one manipulates the fundamental equations of motion in Special Relativity.
This leads to a Third tacit or unwritten Postulate:
iii) to go from one equation in one frame of reference to an equation in the other frame of
reference, the prime variables become un-primed and vice versa and the velocity merely
changes sign.
The Third of Einstein's postulate is evidently a heuristic postulate.
4.0 Einstein's Fourth Postulate of Special Relativity
The functional form of the Lorentz transformation is the simplest function that provides
for a transformation based on the dynamics of the problem. This leads to a fourth
heuristic Postulate:
iv) It is understandable that the Lorentz transformation involves both v and c because
these are the only velocities observers in the two frames of reference can agree upon.
5.0 Deriving the Time Transformation Equation
The derivation of the time transformation involves two steps.
The first step is to eliminate x in the transformation expression for x' , namely
x = y(x- vt) => x = y(y(x + vt')- vt)
The second step involves some basic algebra and simplification and solving for t, which
yields
t = r
t + — X
By Einstein's Third Postulate we find then that
f
t = r
\
t + — X
c J
r
ot =y
\
v
t y x
V c J
6.0 The Relativistic addition of velocities
Using the definition of velocity
Ax_ y(Ax' + vAt')
At
v
\
/I At' + —r Ax'
c
J
/At'
(Ax'
\
V
At'
+ v
J
yAt'\ 1 +
v Ax
'A
c 2 At'
J
(u' + v)
c 2 J
By Einstein's Third Postulate we find then that
(u' + v) , (u—v)
U = -H ^r OU =
1_ 7 M
The value of the third and fourth of Einstein's heuristic postulates are self-evident.
Presented in this fashion, the equations of Special Relativity can be understood by an
individual with a good understanding of algebra.
Acknowledgements:
The author would like to posthumously thank Dr. F.A. Kaempffer and Dr. G.M. Volkoff
for their discussion and encouragement in the writing of this short article. This paper was
originally written in 1984 with their kind assistance. The manuscript remained lost in
the author's papers for several decades, only recently rediscovered and published.
© Patrick Bruskiewich 2103